The appeal of the Golden Ratio to the human eye and brain has been scientifically tested. When subjects are presented with a range of rectangles, people invariably pick out as most pleasing ones those whose sides are of the Golden Ratio. This golden number is the basic number at work in aesthetics; there are even claims that the Golden Ratio was used by Leonardo da Vinci when painting the Mona Lisa, and by the Greeks in building the Parthenon. But the surprising thing is that a number deemed aesthetically pleasing by human beings also crops up in nature and science. In a newly published article in Physical Review B, it is stated that the Golden Ratio appears in the structures of some metals. The Golden Ratio is seen in the arrangement of seeds on flower heads, in the spirals of sea shells and galaxies, even in black holes. This ratio can be found almost everywhere in the universe.

Although the Greek mathematician Euclid first defined the Golden Ratio in around 300 BC, the followers of Pythagoras probably knew of it two centuries earlier. Euclid defined it as a line that can be divided into two unequal parts (Figure 1), where the ratio of the smaller part of the line to the longer part is the same as the ratio of the longer part to the whole. This ratio is 1.6180339887..., the Golden Number.

Figure 1: If you take a Golden Rectangle and take out a square, what remains is another, smaller Golden Rectangle. |

7, 11, 18, 29, 47, 76... When you have written down approximately 20 numbers, calculate the ratio of the last to the penultimate: the answer should approximate the Golden Number.

In mathematical terminology it is (an/an-1) equals to the Golden Ratio.

Figure 2: The ratio of each bone at the top of the hand to the bones at the bottom of the fingers is the GR. |

It was the elusive nature of the Golden Ratio that led the Italian friar and mathematician Luca Pacioli to equate it with the incomprehensibility of God. In the 15th century, he wrote a three-volume treatise, Divina Proportione (Divine Proportion), that was crucial in the dissemination of the Golden Ratio beyond the world of mathematics. After him, many artists, architects, and musicians used the Golden Ratio in their works; for example, musicians such as Debussy and Bartok and the architect Le Corbusier.

Figure 3: Pine-cones show the Golden Ratio spiral clearly. |

The Golden Ratio also crops up in hard sciences. Let’s take a look at the growth of “quasi-crystals.” These maintain a five-fold symmetry, which means that they make a pattern that looks the same when rotated by multiples of one-fifth of 360 degrees. Since the time when these crystals were discovered in 1984, many physicists have been researching their properties. In Brookhaven National Lab in New York State, Tanhong Cai imaged the microscopic terrain of the surface of such crystals made from alloys of aluminum-copper-iron and aluminum-palladium-manganese. It is found that flat terraces are punctuated by abrupt vertical steps. The steps come in two predominant sizes, with the ratio of the heights of these two steps being the Golden Ratio. This fact was discovered in 2002.

Figure 4: A cauliflower has a center point where the florets are smallest, and they are organized in spirals around this center in both directions |

Figure 2 shows the finger bones of a hand. The ratio of each bone at the top of the hand to the bones at the bottom of the fingers is the Golden Ratio. Pine-cones show the Golden Ratio spiral clearly (Figure 3). If one looks carefully at an ordinary cauliflower, one can see a center point where the florets are smallest, and the florets are organized in spirals around this center in both directions (Figure 4). The flower, Echinacea Purpura, has the same spirals (Figure 5).

Figure 5: The flower, Echinacea Purpura, has the same GR spirals. |

*Reference*

• Chown, Marcus, “Why Should Nature Have a Favorite Number,” New Scientist, 21-28 December 2002, 55-56.